# The Illusion of Fit: Spatially Resolved Assessment of Constitutive Model Validity in Elastography and Physics-Based Inverse Problems

arXiv: https://arxiv.org/abs/2502.07415 · HTML: https://mlsanity.com/paper/2502.07415/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 79 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 18 of 18)

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- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty](https://mlsanity.com/paper/2307.02432/) (2023)
- [Conditional score-based diffusion models for solving inverse problems in mechanics](https://mlsanity.com/paper/2406.13154/) (2024)
- [Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastography](https://mlsanity.com/paper/2407.20697/) (2024)
- [Multimodal, high-dimensional, model-based, Bayesian inverse problems with applications in biomechanics](https://mlsanity.com/paper/1512.04481/) (2015)
- [Beyond black-boxes in Bayesian inverse problems and model validation: applications in solid mechanics of elastography](https://mlsanity.com/paper/1803.00930/) (2018)

## Abstract (arXiv)

Inferring the mechanical properties of soft tissues from measured deformations is a fundamental challenge in elastography. A rarely examined assumption underlying existing approaches is that the assumed constitutive law correctly describes the imaged material. When it fails, inversion still yields plausible-looking estimates - an illusion of fit with no indication of local model invalidity, which can mislead clinical interpretation. We propose a probabilistic framework that transforms constitutive model validity from an implicit assumption into an explicit, spatially resolved inference target. The key is to treat the stress field as an independent latent variable rather than deriving it from the constitutive law. This enables a pointwise comparison between the stress required by mechanical equilibrium and the stress predicted by the assumed constitutive model. Both governing equations enter the probabilistic learning objective as virtual observables with separate precision hyperparameters: the conservation law precision is set a priori to a small value reflecting its undisputed validity, while the constitutive precision is inferred under a sparsity-promoting prior. The resulting constitutive precision field provides an uncertainty-aware map of where the assumed model is supported by the data and where it is not. Inference is carried out via stochastic variational inference and is forward-model-free. We validate the framework on synthetic harmonic elastography experiments on a brain-slice geometry with an anisotropic inclusion. The inferred precision field identifies the inclusion with a five-order-of-magnitude precision contrast against the valid domain, robustly across 25-35 dB noise and four-fold sparser observations. A phantom experiment with ultrasound measurements on a linear elastic material yields no false-positive violations and recovers the true stiffness contrast.